You can’t set any page template to the homepage setup through Complexity > Theme Options > Homepage because you’re not actually working with a WordPress page.
http://www.rutherfordhouseschool.co.uk/ — On this link, you’re actually using the homepage builder tool that doesn’t use a “page” at all.
http://www.rutherfordhouseschool.co.uk/homepage/ — And on this link, you’re using a standard page that you’ve named “homepage”.
If you want that page called “homepage” to be your front page then you need to go to Settings > Reading > Frontpage displays, select a “static page” and select your “homepage” page you want to be the front page.
However, by doing this, you won’t be able to use the theme’s homepage builder. You’ll just be creating a regular page like with any other WordPress theme.